Could you please explain how to find the slope using the definition of derivative of the function y=2*sqrt(x)
http://tutorial.math.lamar.edu/Classes/CalcI/DefnOfDerivative.aspx
Certainly! To find the slope using the definition of the derivative, we need to find the limit of the average rate of change of the function as the interval over which we calculate the rate of change approaches zero.
The average rate of change over an interval [a, a+h] is given by the formula:
average rate of change = (f(a+h) - f(a)) / (a+h - a)
In this case, the function is y = 2√x. Let's calculate the average rate of change using this formula.
First, substitute f(a+h) and f(a) into the formula:
average rate of change = (2√(a+h) - 2√a) / (a+h - a)
Simplify the expression:
average rate of change = (2√(a+h) - 2√a) / h
Now, we want to find the limit of this expression as h approaches zero. To do that, let's simplify further.
Multiply both the numerator and denominator by the conjugate of the numerator, which is (2√(a+h) + 2√a):
average rate of change = [(2√(a+h) - 2√a) * (2√(a+h) + 2√a)] / (h * (2√(a+h) + 2√a))
Using the difference of squares formula, simplify the numerator:
average rate of change = [4(a+h) - 4a] / (h * (2√(a+h) + 2√a))
Simplify further:
average rate of change = 4h / (h * (2√(a+h) + 2√a))
Cancel out the h terms:
average rate of change = 4 / (2√(a+h) + 2√a)
Finally, take the limit as h approaches zero:
limit as h approaches 0 of (4 / (2√(a+h) + 2√a)) = 4 / (2√a + 2√a)
Simplify the expression:
limit as h approaches 0 of (4 / (2√a + 2√a)) = 2 / √a
Therefore, the slope of the function y = 2√x using the definition of the derivative is given by 2 / √a.